Question

Two different dice are tossed together. Find the sum of the probabilities
(i) that the number on each die is even.
(ii) that the sum of numbers appearing on the two dice is $5$.

Answer 1

Let S be the sample space.

$S=\left\{\right(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(2,1),(2,2),(2,3),(2,4),(2,5),(2,6),(3,1),(3,2),(3,3),(3,4),(3,5),(3,6),(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),(5,1),(5,2),(5,3),(5,4),(5,5),(5,6),(6,1),(6,2),(6,3),(6,4),(6,5),(6,6\left)\right\}$

(i) Let A be the event that the number on each die is even.

$\therefore A=\left\{\right(2,2),(2,4),(2,6),(4,2),(4,4),(4,6),(6,2),(6,4),(6,6\left)\right\}$

$\therefore n\left(A\right)=9$

$\therefore P\left(A\right)=\frac{n\left(A\right)}{n\left(S\right)}=\frac{9}{36}=\frac{1}{4}=0.25$

(ii) Let B be the event that the sum of numbers appearing on two dice is 5.

$B=(1,4),(2,3),(3,2),(4,1)$

$\therefore n\left(B\right)=4$

$\therefore P\left(B\right)=\frac{n\left(B\right)}{n\left(S\right)}=\frac{4}{36}=\frac{1}{9}=0.11$

Hence, the sum of the probabilities is $0.25+0.11=0.36$.